I have run RNAs out on a standard EtBr agarose gel - you can see the rRNA bands if the RNA is concentrated enough and intact. If the RNA is good there will be minimal smearing and obvious rRNA bands.

I agree with Amanda and Robert. The easiest, if you don't have an analyzer, is the use of agarose gel electrophoresis. A standard TBE or TAE gel is fine. As 28S is ~2x the size of 18S, it should be twice as bright. I have not seen a need for a denaturing gel. You can always run a very thin gel if you don't have a lot of sample or if you want to run a quick analysis.

Run agarose gel to check integrity of rRNA.

Why do you not want to use a Bioanalyzer or Experion or a similar instrument? This is the gold standard and there are plenty of service providers that you can send the samples to. One of them is Micromon at your university.

Without more details it is hard to troubleshoot, but I can give you my protocol:

I make bleach gels with 1% bleach v/v, 1% agarose w/v, 0.01% v/v SYBR Safe in 1X TBE, and run 1ug RNA from a variety of mouse organs for 15min at 100V in 1X TBE.  

I isolate RNA with TRIzol (1ml per 50mg tissue, double for pancreas) typically on whole organs, or in the attached image, 30mg. Phase separate (phase lock gel makes this much faster and easier) with 1/5 volume of chloroform per volume TRIzol. I've found that mixing the aqueous phase with a half volume of isopropanol and transferring to an RNeasy Mini column gives the cleanest RNA with identical integrity, although your total yield will be a little less compared to manually pelleting and washing. After spinning through RNA/isopropanol mixture, 700ul RW1 --> 500ul RPE --> 500ul RPE 1min. Transfer to new collection tube, 1min centrifuge. Elute with an appropriate volume of water based on your sample (incubate water on the membrane for 5min prior to spinning to improve yield). Spin through and optional re-elute with eluate to increase RNA concentration. I found that incubating resuspended RNA at 42C for 15min evaporates any residual EtOH, although this should generally not be required (I used this for white adipose tissue RNA, which was floating out of my gel well).