I am trying to trace my fathers side of the family back to Russia. Unfortunately, I cannot seem to locate my great grandmother, Anna Alexonis' family which remained in Russia. Does anyone know of a legitimate Russian database that I could search? Thank you.

Alexonis is a Lithuanian name. The family probably comes from what is Lithuania today. Some chances are they came from a different part of the Russian empire as well.
It does not make much difference at this point, as no database of the kind you are asking for exists. Also, the name is of medium incidence and you may start to chase a wrong track even if you will find a similar name somewhere.

What you need, is to find more detailed information at "your side" about where from and when your ancestor did come in; and then look for a possible appropriate source at the "other side".

Thank you for your response. I have made contact with someone at the Lithuania State Historical Archives who may be able to assist me with my search.

Yes, that's what I had in mind.

However, as far as I know, the for a standard inquiry you need to provide a name (which you have), a location or at least a hint which church it may have been, and an estimated date, then archives will search within +/-5 years.

I have recently heard back from the Lithuanian State Historical Archive. I had sent an email to them about a month ago re: my great grandparents Lithuanian/Russian roots. They tell me that they have possibly found my grandfathers birth name in the birth registry book kept by the Liskiava Roman Catholic Church. I hope that 2013 brings me closer to knowing my family roots!

I'm curious as to whether you provided the Archives with a location to check records for.

There is no real index of births/deaths for the period of time you are interested in and you generally need to give them the name of the church that you want them to search.

Tom S.