Actually, connecting to the case is probably better unless you also connect the case to earth ground. The easiest way to ground the case is to plug it in, but you don't want to do that until you're ready to power on.
What you're trying to prevent is Electrostatic Discharge (ESD), also known as a static spark, from you to the part(s) that you're working with. Nothing in that equation involves earth ground except that other things that you're likely to touch might be grounded.
So when you're working on stuff, you don't really need to be earth grounded any more than when you're not working on stuff. You just need to be connected to the stuff that you're working on.
An ESD mat will be conductive enough that you can just set the case on it, put the strap on, and be sufficiently connected to the case. Then handle all the other parts by whatever it is that will eventually touch the case anyway, and you'll be fine. No need for earth ground in most cases, though it usually doesn't hurt either.
As a side note, I've built several PC's on carpet without any grounding at all just by handling parts as described above and by casually touching the case a lot. Never had any problems. But I'm also naturally paranoid enough in general to remember to do that.
The 1meg resistor is needed to safeguard the user from faults from other equipment connected to mains earth.
Keep in mind that the wrist strap is a permanent connection to the electric system of the building. If another piece of equipment experiences a fault, there could be a big fault current through the mains earth wiring system. That means that, in unfortunate circumstances, the mains earth terminal could reach a dangerous potential. In this case, the 1Meg resistor limits the current from the earth wire trough the user to a safe limit.
See this Wikipedia article about Earth Potential Rise, for example.
Excerpt:
The resistance of the Earth is non-zero, so current injected into the earth at the grounding electrode produces a potential rise with respect to a distant reference point. The resulting potential rise can cause hazardous voltage, many hundreds of meters away from the actual fault location.
Therefore the earth wiring system (and your wrist), due to its low resistance, is roughly at the same potential of the point where the fault current enters ground, whereas your feet (several hundred meters away from that point) are at a lower potential. Without that 1Meg resistor: ZAPP!!!
EDIT (to address downvoting and clarify my answer)
Since my answer has attracted a couple of down-votes and some criticism in the comments (not necessarily related, at least not apparently) I feel compelled to clarify something, but I'd like also to remind downvoters what down votes are for: for answers that are not useful, not on topic or plainly wrong.
First: I was told that regulation doesn't require the 1Meg resistor for the reasons I stated. My answer: I never stated that my explanation was related to some regulation (I didn't even know there was a specific regulation for wrist bands - BTW, I'd like to see some reference), but I concede I could have been more explicit.
Second: As I wrote in a comment, I admit that my scenario is less likely than, for example, touching a live wire or an ESD event whose rapid discharge could cause issues. Nevertheless, as someone said in a comment, You only die once! Faults in electric systems do happen, and often they are not under your control, so no level of care from your side could prevent them, you can only (try to) prevent the consequences. Hence the scenario I depicted is, IMO, well worth considering (so it is on-topic and it is useful). Moreover, the question in the title is Should there really be 1 MΩ resistance between an anti-static wrist strap and a pc?, not something like Why regulations impose a resistor there? or What's the most likely scenario that resistor is put there for?.
To further make my point you can see this article on Wikipedia about Stray Voltages. Not everything is directly related to what I'm saying, but the part on Neutral return currents through the ground is. Excerpt (emphasis mine):
Stray voltage became a problem for the dairy industry some time after electric milking machines were introduced, and large numbers of animals were simultaneously in contact with metal objects grounded to the electric distribution system and the earth. Numerous studies document the causes,[11] physiological effects,[12] and prevention,[13][14] of stray voltage in the farm environment. Today, stray voltage on farms is regulated by state governments and controlled by the design of equipotential planes in areas where livestock eat, drink or give milk. Commercially available neutral isolators also prevent elevated potentials on the utility system neutral from raising the voltage of farm neutral or ground wires.
(I didn't have the time to search for an article involving grounded humans instead of grounded cows, but you get the point.)
Bottom line: connecting a human body to any low-impedance path that could possibly rise in potential is dangerous and life threatening, so proper safety measures should be in place.
Best Answer
Antistatic precautions mean you should never move a device through a large potential difference.
Once you and the ESD mat are connected to ground, they are at the same 0v potential. Take hold of the bag, this bring it and its contents to 0v. Now you can open the bag and handle the devices with anything that's also at 0v. This includes fingers, or conductive things you are holding. So metal tweezers, 'antistatic' pliers that have special conductive plastic handle covers, cotton gloves (which retain enough moisture to be slightly conductive) but NOT tools with insulating handle covers or plastic gloves.