Easy to tell. Where is the rear sight? Is it in front of the chamber like a mauser or over top of the bolt handle, with a peep-sight? If its a peep type, its an 1903A3 and is generaly exempt from the heat treatment issue all togather. If its a 'stanard' sight then you need to have it run as it could be a WWI or pre.
Just an anecdote, not ment to frighten you off, but nearly all 1903s were produced as "2nd line" weapons, or "budget rifles". During WWI they were never issued in large numbers, being that 10 M1917 Enfields were made in the US for every 1 M1903, and in the '20s the US Army was on such a shoestring budget it often cut corners and the quality of the workmanship on the 1903s suffered. By the time WWII rolled around, 1903s were playing second fiddle to Garands and Thompsons in production and are, even compared to wartime 1903s and 03A1s, pretty rough, like the difference between a stamped and a milled SKS. My only 1903 is a 1943 Smith Corona A3 and it did let go on me (sort of) after shooting a couple of hundred rounds of whatever swill Century sold me, I developed a stress crack in the threaded area of the reciever around the barrel and is no longer safe to shoot. This has happened to a few other WWII production A3s that I know of, but not on any sort of grand scale. I know of just as many 1903s that have had THOUSANDS of rounds put through them and dont even look the worse for ware. If your grandpa shot it a lot "in the day", my money is on yours being perfectly safe, but as the guys have already said, take it to a good 'smith who is competent with milspec rifles. One of my buddies just took a 1908 Short Magazine Lee Enfield with all the original goodies on the 1st run SMLEs, to have it checked out, and his smith ended up microbedding the rifle and ruining its collectors value and its shooting charactaristics.
That said, enjoy one of the few great American icons WE ripped off from someone else (instead of the way it ususally is). The 1903 is a very interesting bastard brother to the Gew98 Mauser (actually its more kin to the Spanish 95 that we faced in Cuba). There is SO much to know about them...