Applying heavy pressure is good and all but against someone who can pick up the heat it simply is going to leave you getting gashed due to the receiving core and because Rodgers may be the best QB at evading pressure and getting off an accurate throw in the game.
Now with that said I wouldn't want to leave Rodgers with time to throw either (at least not on a consistent basis) as you'll get eaten alive that way too.
Being someone who plays as the Packers I'd attack myself with a combination of man and zone coverage (along with combination coverages) so that I'd have to slowly move down the field while throwing in the occasional blitz (a good zone blitz would be ideal as press man blitzes will just get beat deep). This would also help confuse the pre-snap reads which will instill some doubt meaning you have a better chance at a mistake occurring. If you constantly blitz you will get picked apart but if you bring them at random intervals you'll be more likely to cause confusion and thus get to Rodgers meaning that you'll be much more likely to have a drive stall out. I also advise to stop the running game. Even though their backs aren't elite Grant and Starks can be the death of you if you let them pound off a 3.5+ ypc for the game as it opens up so many other plays to get Finely involved in.
You're goal shouldn't be to completely shut them down (if they are being played by a good player) but instead you want to force them to slowly march up the field and try to force some mistakes during that long drive. The Green Bay offense isn't going to be shut down if they know what they are doing but playing an opportunistic defense is your best bet.