It should be OK to plug things in, BUT there are a few things that could be small problems.
1. Plugging a 3-prong plug into a 2-slot outlet requires an adapter, and most of these can be plugged in both ways - "right side up" or other way. What's the "right side"? Well, that is not easy to tell, but a voltmeter can help. To start, think about a 3-prong outlet. Imagine it installed so that the round Ground hole is below the other two flat slots - that is, so the Ground is at the BOTTOM of the triangle of holes. In that orientation, the Neutral line is the slot on the LEFT, and if you look closely you'll see that it is wider than the other slot. The Hot slot is the one on the right. With a 3-prong plug, you can't get those two reversed. BUT when you insert the adapter, you could, because most adapters do NOT have their two blades of different widths, and they'll go in either way. Now, on MANY devices this would not matter, but some items depend on having the Neutral blade of their plug inserted into the Neutral slot of the outlet, etc. If you want to be sure, get a voltmeter and a long piece of wire. Attach the wire to a known Ground, like a water pipe as close to the pipe entrance point as you can. (The reason for this is to be sure you do have a Ground, and are not isolated by some intervening piece of plastic piping.) Run the other end to near the 2-slot outlet you plan to use. Now measure the AC voltage between this ground lead and each of the two slots in the outlet. One should be at 0 volts, and that is the Neutral line. The other should be at 110 volts AC, and that is the Hot line. (This all works because in North America, the common system is that the Neutral bus at the fuse panel is connected to earth Ground - it's called the Grounded Neutral system. BUT this does NOT mean that the Neutral line can be used as a Ground - it CANNOT be relied upon to be at exactly zero volts because it is carrying currents.) So now you know for sure which slot of the 2-slot outlet is Hot (hopefully on the right, but maybe not), and you can plug in your 3-prong-to-2-prong adapter correctly.
2. Even on the devices that have only 2 blades on their plugs, check whether they are the same width or different. Many newer devices have them different so that they can only plug into a 3-prong outlet properly, even though they are not using the round Ground hole connection. Some, however, do not and both blades are the same. This is important because it is VERY likely that your 2-slot outlets do NOT have one slot wider, and they will not accept a plug with one wide blade.
If you're having the whole house re-wired, that is the best solution to this situation. But temporarily things should work. Having proper Grounded 3-prong outlets is a big safety feature for people, and to some extent for electronics. But one real advantage to them is that the electronic devices actually have a connection to a true earth Ground for purposes of shielding out stray electronic noise signals. That can prevent the odd signal malfunction or slow performance of equipment.
When you or your electrician are planning the re-wiring, make sure to consider the power requirements of your computer system. I installed one completely separate circuit (15 amps, 110 VAC, 4 outlets plus using a multi-outlet Surge Protector) to my computer area so it is not "competing" with other devices for power and is all fed from a single voltage line in the breaker box.