I see no reason why I(1) varibles would not be appropriate. Potential bial would depend on specifics.

The problem with multiple I(1) time series can be shown as follows. Let there be two independent time series, x_t and y_t. They are both an I(1) random walk, as follows:

x_t = x_t-1 + errx_t

with zero mean errx_t. And similarly for y_t. Let the regression model be:

y_t = a + b x_t + residual_t

From the independency follows that the true b is zero. However, "the estimated b parameter will seem significant with a large absolute t-ratio and also the R2 of this regression will be close to unity, see Granger and Newbold (1974) for simulation evidence and Philips (1986) for a formal statistical treatment". (Quoted text from Franses, "Time series models for business and economic forecasting" (1998), page 213/4.)

More general, let both series be I(1) with drift:

x_t = x_t-1 + driftx + errx_t

and similarly for x. If for instance both drifts are positive, then one might expect two gradually increasing series, with ups and downs of course, giving the above described wrong regression result, in this case with positive estimated b.

Search for "spurious regression" and "cointegration".

If the explaining variable(s) is I(1), then the explained variable y_t = a + b*x_t will be i(1), too. If all err's in Arie's example are symmetrically distributed with mean zero, the residual_t will also have a mean of zero, the variance will be somewhat different. An OLS-estation will likely bring correct results. But I fear, that x_t = x_t-1 + errx_t does not mean, that x is I(1). For I(1), the error term must not be symmetrically distibuted around zero, thereby producing trends or cycles. Therefore, the results of a regression will likely be wrong (spurious; see Arie's drift example). One should try transformations (e.g. differences) to get rid of I(i).