If you can, I would look underneath the seating area. Follow the levers and bellcranks of the shift linkage down to the transmission case. Try to disconnect the linkage from the transmission selector. Do not remove the lever on the side of the transmission. Some will allow the mechanism to fall out of place if removed.
Try to move the linkage separate from the transmission. If it is binding, figure out where, and count yourself lucky.
If it moves relatively freely, you likely have a frozen/rusted transmission selector. The lever on the side can bind due to electrolysis between the lever & case metals, or just plain corrosion. To free it up, you have to determine if it is the external lever or the internal detent locator, or the linkage the moves the internal forks, or the forks themself.
There is a series of posts in another thread that shows some internal parts so you have an idea in general of what goes on to change gears. If you are up to it, and don't have $$'s for a replacement {craigslist..ebay...}, you can remove it and spit the case, inspect, clean, de-rust, free up, and so on. What do you have to lose? Unless you like running in 3rd all the time. Which is a viable plan for most mowing, just don't get caught in a corner where you have to back up.
tom

 

If your real lucky,it could just be the shifter mechanism is frozen from rust,etc..

I'd be willing to bet if its one of the aluminum cased transaxles that used a "straight line" 5 or 6 speed shifter,there usually is no "linkage",the shifter goe into the case itself..it "shifts" by sliding the internal gears or "dogs" on a keyed shaft ,and if it has had enough condensation form inside it after sitting,to make a nice puddle of water for the gears & shafts to bathe in, and rust,it'll be "stuck"..

This of course will make it hard for the dogs or gears to slide on the keyed shaft,and it'll be "stuck" in whatever gear it was left in before sitting so long..sometimes the shift keys get burred or worn too..(search for "shift keys" threads on this forum and you'll find some threads with pictures that will help you understand what I'm talking about)..

If you get it to run,you can try jacking the rear wheels off the ground and let them spin in gear awhile,that may help to free the gears up and get the old grease that is probably hardened up and settled to the bottom to spread around--if your lucky you might be able to get it to shift while it is running "in gear" and free itself up..
It might require the transaxle to be pulled out and taken apart,cleaned up inside,and new lube installed,if its really "stuck"..

 

Thanks for the suggestions.

Tractor-Holic, as soon as I get it running I will let the wheels spin and cross my fingers.

tomw0, if it's always in 3rd gear it will be moving whenever the engine is running? Or will the clutch/brake pedal work like a car clutch? I clearly have 0 experience with riding mowers.

Thanks again.