I do most of my flying in the mountains on the French / Italian border.
Flying through the valleys from France to Italy is a perfectly feasible thing to do, and the only problem with doing it in a PA28 would be the climb performance if you were heavily loaded on a warm August day.
But...... there are lots of things to be very aware of, and it's not advisable to try to navigate through the mountains if you aren't very familiar with them.
First, map reading. A standard 1:500,000 chart is completely useless in the Alps. I've got to know my local area fairly well, and don't use a map. If I am planning a flight outside my normal area, I study the 1:100,000 chart in detail on the ground, work out where the valleys I want are in relation to landmarks, and try to figure out where the cables are likely to be.
Second, cables. The alpine valleys are full of cables, and only a few of them are marked on maps. Below the treeline, there are numerous cables left from current or past logging operations. Above the treeline there are cables used to transport explosives used for causing small avalanches. One valley near the Italian border just South West of Mont Blanc has 6 of these cables in a very small area, and they aren't marked anywhere. They are several hundred feet above ground level.
Third, navigation error. If you turn up the wrong valley, it can spoil your day. You always need to be aware of how much space you need to make a 180 degree turn, and you need to make that turn as soon as you are in any doubt about your position.
Fourth, weather. In mid summer it can be extremely turbulent, and the myth about the downdrafts not going all the way to the ground is just that - a myth. I know a PA28 pilot and a glider pilot who both crashed because of this. You need to be aware of the upper winds, and the likely local wind / thermal effects. It's quite feasible to fly, and if you use the thermals well you can have a lot of fun - but you need to know what to expect, and your passengers may need sick bags. You can also expect a lot of cumulus build up in the afternoon, which makes identification of landmarks difficult, and you can expect CB development on many / most warm afternoons. You really want to be on the ground then.
Fifth, airspace. There are numerous nature reserves which you mustn't overfly at less than 1000ft AGL, and you need to know where they are. There are also national parks which have a 3000ft AGL limit.
I don't want to put you off - there's no more exhilarating flying than mountain flying. But do be forewarned, and preferably have a flight with a mountain instructor. In a couple of hours flying he will be able to make you very aware of all the major hazards, and you'll be able to enjoy the experience a lot more.