In reply to LondonClimber:
I've done a couple of short trips - nothing technical but we took harnesses, rope, axes and crampons.
Trip 1 - Zakopane and Tatras national park. Zakopane is (I think) the main skiing town in Poland so is relatively prosperous and expensive. The area has a lot of famous routes and mountains. Potentially serious in feel although we were just hiking. -10 on the tops. Hiked up to a hut and stayed a couple of nights. A couple of hours up to the hut - not too strenous. The walks to the ridges needed crampons and axes. I was glad we had a couple of local friends who knew the area.
Trip 2 - We 'did' Sněžka on the Polish/Czech border. We stayed the first night in Karpacz and then hiked up to this place - http://strzechaakademicka.pl/ and then up to the top and down the next day. We walked past a couple of other huts - all looked fine with decent accommodation and good hearty food. We had the clothing and gear in case the weather got bad but there were a lot of folk about who looked like Ben Nevis summer tourists. We took it easy and had nice, pleasantly tiring days. You could go much harder or do shorter 'up and down' routes. It's popular and busy so doesn't feel too commiting but as you say, in bad weather and visibility it could get challenging quickly. All hiking and no climbing.
Vang Stave Church in Karpacz is worth looking at (wooden Norwegian church moved to Karpacz). Lots of places to stay and eay in Karpacz.
Let me know if you want pics to get an idea of the place.