As with all engines, you need spark, fuel and compression. If all three are present it will run, if any are missing it won't.
When you try to start the engine, does it turn over at normal speed, as if it is trying to start, or does it spin a lot faster or slower than you might expect? If it is a lot faster, you may have low compression. If it is a lot slower, your starter motor is probably suspect (assuming it has a starter motor and isn't a pull-start like smaller mower engines).
Assuming you have good compression and the engine is turning at a good speed, the next thing to check is the spark. Take one plug out and check its colour against a colour chart*- this can often tell you a lot about what is going on. At this point, it is usually wise to check the plug gaps - you'll need a feeler gauge and the mower's workshop manual for the correct values. If the plug looks good, connect the plug to its lead, and lay it on top the engine so the body of the plug is earthed, then try starting and watch the plug - you should see a spark. If you don't, or it looks weak, that is the problem. If you get a good spark, replace the plug, re-gap the others, and move on to fuel...
Unless it is gravity-fed, there will have to be some kind of fuel pump somewhere in the system. A quick way to test this is to get a can of "easy-start" or similar. Remove the air filter, and (working in a well ventilated area) spray the easy-start directly into the intake, then try to start the engine - if it coughs, then fuel supply is the likely problem. If you've checked the fuel filter, then it is either a failed pump or a blockage in the line. Disconnect the inlet to the pump, and pop the end of it into a can of fuel (being careful, petrol vapour is highly flammable!), then try to start - being in mind will take longer than normal to suck the fuel through. If that works, it is a blockage in the line (or the fuel in the tank is crap). If not, remove the pump and test it off the engine.
*This example being the first returned by a google image search.