triangular grids in general lead to good interpolations but are more time consuming

take a look at this article

The design of optimal sampling schemes for local estimation and mapping of regionalized variables—II ☆: Program and examples

A.B McBratney & R Webster

http://www.sciencedirect.com/science/article/pii/0098300481900789

There are always issues regarding the accuracy when interpolation is done. I have found krigging to be quite realistic interpolation. It will surely depend on the grid (mesh size) you are considering. But I have not faced any serious issue interpolating data both in air and marine waters using krigging. 

Choice of interpolation method  does depend upon the phenomena to be studied and of course accuracy required etc. Please see the file attached.

Kriging can be carried out on a triangle grid successfully. In fact, accuracy of an interpolation depends on an appropriate density of sampling points and sampling strategy. In addition nature of surface (cnontinuity and discontinuity) as well as homgeneity of surface and  accuracy of the data are also have certain bearing on the accuracy of interpolation results as it can results in the interpolation biases and deviation from reality.

Respected friend use geosptial analysis to make map of sample data. good luck

I am of the opinion that such sampling technique will lead to lower accuracy level as a continuous surface made from triangles as compared to squares is most likely going to have higher accuracy clustered at some point within the triangle, instead of evenly as with squares. Also it's almost impossible to cover a regular area with triangles of equal spatial resolution, not unless the area is a triangle itself, and we know that in reality areas are not bounded as triangles.

Deborah, the problem you have suggested will be same with squares. To have a similar accuracy with squares you will have to make much denser grid which means much longer computational time. two equilateral tringles one upright and another inverted makes a sqaure, so an accuracy assessment and you will see that tringular grid is much accurate.

triangular grids in general lead to good interpolations but are more time consuming

take a look at this article

The design of optimal sampling schemes for local estimation and mapping of regionalized variables—II ☆: Program and examples

A.B McBratney & R Webster

http://www.sciencedirect.com/science/article/pii/0098300481900789

Alejandro is correct....I think Yfantis 1987 also published related results.

http://link.springer.com/article/10.1007/BF00897746#page-1

Kriging is Geostatitics, and use variogram to fix the spatial model

Three issues need clarification: 1.Do you mean that the region enclosing the data locations is triangular? If so, the shape of the region is not relevant to the interpolation problem. 2. You didn't say what kind of data you have, i.e. what quantity is being measured or recorded. Do you mean that the measurements are made on triangular subregions, e.g. average values over triangular subregions? If that is situation then you are talking about non-point support and that is an entirely different question. 3. Alternatively maybe you are saying that the data locations are the vertices of triangles (but point support data). Both with respect to any subsequent kriging and also the preliminary step of estimating and fitting a variogram a triangular pattern for the data locations does not have any particular advantages. An irregular pattern is actually better for the variogram estimation/modeling step.

If your data is averages over triangularly shaped subregions then you will have to customize the software for kriging and also for variogram estimation and modeling, all the software I know of assumes rectangular or cylindrical supports. Re-programming for triangular supports will be quite difficult.

If you have non-point support data then the old book "Mining Geostatistics" by A. Journel and Ch Huijbregts is still a good reference

Donald, thank you indeed for your answer and for making  the question much more clear! I add some details, as suggested, following your points:

1. Yes, the study area is (more or less) triangular.

2. My data will be basically metal concentrations bioaccumulated in lichens which are going to be transplanted on trees at given sampling sites. And

3. Yes, I mean that my sampling sites will be possibly allocated at the vertices of a triangular grid.

So, if I got rightly the message: this possible design (triangular area with sampling sites at the vertices of a triangular grid) has no relevant problem, but (especially for point allocation) no particular advantages. As for the allocation, an irregular pattern (e.g. the one that could come from a simple random sampling) could be even better. Isn't it?

"Both with respect to any subsequent kriging and also the preliminary step of estimating and fitting a variogram a triangular pattern for the data locations does not have any particular advantages"

Not quite right...see the papers referenced in previous comments.  Use of a triangular grid minimizes estimation variance over the space of unsampled locations for a fixed number of sampling locations.  Triangular is better than rectangular or irregular because the maximum distance to supporting points is minimized...per the Yfantis paper.

With a primary objective of minimizing estimation error, variogram estimation becomes a secondary (and competing) objective that can be accomplished by doing three things.

1) Lay out the triangular grid at a spacing using approximately 80% of your available resources.

2) Randomly select a small number (10%-20%) of grid nodes at which a field duplicate will be collected within a short distance at a random distance and direction.  This is for estimating nugget effect of the variogram. Irregular and systematic approaches will not provide the necessary data to estimate this critical parameter.  All uncertainty estimates are highly sensitive to proper estimation of the nugget. 

3) if there is an indication that the triangular grid spacing is more than about 1/3 the expected range of influence  allocate a another 0%-10% (depending on step 2) of effort to to another set of randomized nodes at which an additional sample is selected again in a random distance and direction from the selected node and at a maximum distance of about 1/2 the triangular node spacing.  You can get a feel for the likely range of influence by simply observing the area and estimating the size of patches of lichens.  Patch size is generally about twice the range of influence.   

This approach will achieve the primary objective of developing an accurate map with most uniform control of estimation error possible, while also providing the auxiliary data needed to obtain a quality estimate of the key range of influence and nugget effect.

1-3 are in prioritized order from most important to least important.  

Whenever you use kriging to interpolate spatial data there are two distinct steps. First you have to estimate/model the variogram/covariance function and then secondly use that spatial function in the kriging equations. It is certainly true that in the second step that having data locations close to the interpolation location is an advantage, in particular the kriging std deviation will be smaller. But in the first step you want to have a mix of separation distances (between data locations) and to achieve that with a triangular pattern requires many more data locations and closer together (smaller triangles). Hence the question is how much does it cost to collect data at more locations and to lay out an exact triangular grid? In most cases there will be compromise.