To attempt to get a very, very inexact answer, I went back and combed through the Patriots drafts since 2003. I'm automatically eliminating positions of strength (linebacker, tight end) and focusing on positions Dimitroff will likely be charged with improving (O-line, safety). Rather than project out each specific pick, I'll look at what we'll most likely be targeting in each round under the good Comrade.
Round 1, QB: I'm going to run contrary to history immediately and say the team picks a QB here. It is possible that Dimitroff will dismiss the QBs at the top of the draft, but pressure from ownership and the team's glaring need will likely rule the day. In the past five years as head of college scouting, Dimitroff has selected two defensive linemen here (Ty Warren and Vince Wilfork), a tight end (Ben Watson), an offensive lineman (Logan Mankins), a running back (Laurence Maroney), and a safety (Brandon Meriweather). If Dimitroff is thinking outside the box here, my guess is that he'll try to grab an offensive lineman to improve another of the team's biggest weaknesses.
Round 2, OL/S/CB: This holds true for Round 3, as well. Dimitroff has never been shy about picking up safeties in the high to middle rounds of the draft, and the Falcons will have to decide if they think Daren Stone is a viable option once Lawyer Milloy shuffles his way to the locker room for the last time. Offensive lineman is a huge need and is guaranteed to be addressed more than once; ideally Dimitroff will look hard at stocking depth at tackle and guard. Cornerback is a need as well, though whether DeAngelo Hall stays or goes and whether David Irons improves should determine how high the team is looking. History tells us Dimitroff will alternate offense and defense here, so look for a lineman in the second round and a safety in the third.
Fourth Round, DT/RB: I think a lot of Trey Lewis. Still, Dimitroff is likely to want a versatile defensive tackle who can provide some inside pressure. That's especially true if he's not sold on Jamaal Anderson, who couldn't find the QB with a map this season. I don't think he'll be ready to replace Anderson right yet, but with Coleman not getting any younger and some question marks at the position, this is where I could see Dimitroff addressing it. There's also a possibility that he'll try to draft and groom a running back to work with Norwood and Snelling.
Fifth Round, K: Everyone loves Morten Andersen. Without him, my days would be spent actually thinking up original jokes about other players on the roster. Still, I don't think anyone's going to argue that the Falcons could use a long-term solution here. Dimitroff pulled the trigger on Stephen Gostkowski in the fourth round two years ago, showing he's not gun shy about grabbing a leg attached to a person if they're a good fit for the team. I think this is the year the team finally decides they need a kicker and lets Andersen take his amazing act on the road.
Sixth Round, WR: Still a need. Joe Horn is old, Jenkins is inconsistent and Adam Jennings looked mostly putrid all season. Even if Laurent Robinson emerges, this team could use a capable slot receiver with some room to grow. Enter Dimitroff, who was on board in 2002 when the team selected the temporarily effective David Givens in the low rounds. The Falcons have spent too many high-round picks on receiver the last few years and should instead be looking for a steal here. Whoever they get can't possibly be worse than Horn, anyways.
Seventh Round, QB/OL/CB: As you can tell, I'm just taking a shot in the dark here. The Falcons may want another young QB to groom for the backup role if Redman stays. They might also want to continue to improve the OL depth. They might even go for a fourth CB/KR here, just to replace the beloved (but useless in coverage) Allen Rossum. Dimitroff also may try to add depth at a position I haven't even mentioned, such as LB or DE.
I admit I'm projecting a little bit of my own view of team needs here, but I do think Dimitroff will come to many of the same conclusions. If you don't agree, tell mere where you think the Comrade will look.