You may be eligible for a conviction reduction or possibly even an expungement depending on what the charge was. In Utah, you only have to wait 7 years to have a third degree felony expunged from your record, assuming you are otherwise eligible. I suggest you visit http://www.crimebegone.com and take the eligibility test. If you are eligible for an expungement, you may want to consider that first as opposed to just reducing the charge.
The answer depends in part on what you mean by "did my time." Under Utah Code 76-3-402, a person who successfully completes probation can be eligible to have their charges reduced one or two steps.
So, if by "did my time" you mean you went to prison and completed your sentence, then you would not be eligible for that 402 reduction. If you mean that you successfully did your time on probation, then you may be eligible.
I agree with the answers above regarding the possibilities for reduction of your felony to a misdemeanor. The final decision as to whether it will be reduce will be up to the judge. With the prosecutor's approval, it could be reduced by two sentencing levels. If it has been over 10 years, and you have an otherwise clean record, you may be able to get the conviction expunged. You should consult with a lawyer, however, regarding wether an expunged felony will make you eligible to possess a firearm under current federal law.