In Ecuador, the highest one without glacier travel would be Illiniza Norte (5105m). Depending on conditions, you may need crampons. There are many peaks in the 4000-5000m range as well.
In Peru, January and Februari are not the best months to climb high mountains: this is the wet season. You'll have to go to the south for the highest ones without glaciers, and Chachani (6057m) and Misti (5822m) are your best bets. Misti especially is popular with backpackers, Chachani is climbed less often. Both are basically just walk ups and are climbed year round, albeit less in the wet season. You'll need crampons. That said, January and February are not the best months for it, as it is the rainy season. Fortunately, that doesn't mean it rains (or snows) all the time, just more than in the dry season, so if you have a bit of time to wait for good weather, you'll still have a good chance to make it. But the likelihood of a nice summit view is definitely lower than in the dry season. While waiting, you can get acclimatized by visiting Cuzco, Machu Pichu and lake Titicaca, and don't forget to enjoy the great hiking opportunities in the Colca Cañon and the area around Chivay, where there are several peaks over 5000m, with and without glaciers. Back in May 2011, while acclimatizing, one of the nicest peaks I went up was Huarancante (5426m). No glacier, but I needed crampons.
Bolivia I don't know well enough to give solid advice, only a possibility. The highest peak that can be climbed without glacier travel is Parinacota (6342m). It's mostly climbed from May through October, because it's almost guaranteed to be dry then. The main problem in that period is that you can occasionally get strong winds, more likely late in the season, and that, combined with the low temperatures, can make it dangerous. I climbed it in September 2009, late in the dry season, and while there was still a bit of old snow here and there, we didn't need crampons. As with southern Peru, January and February are the rainy season. but as the area is basically a high altitude desert, I wouldn't rule out the possibility of a summer ascent. Perhaps somebody else can say more about that.
That time of year, the only ones in Peru I'd suggest are the peaks around Arequipa as mentioned above.
El Misti, Chachani, Pichu Pichu, Ubinas (if it's not exploding), etc. It is the rainy season, but that part of Peru is the desert and it's possible to still possible to find good weather (expect some bad weather at times too).
Here is my trip report from the rainy season (but November is not quite as wet as January/February):
Same as Peru. The only areas I'd head to are the desert ones. Licancabur and the surrounding peaks are reasonable then. In fact in the isolated areas in that region, the wet season is probably easier for climbing as there is usually a little snow around for water (during the dry season, water can be a big problem).
Ecuador is more heavily glaciated than the desert areas mentioned above, so you won't get as high without glacier travel. The only 5000 meter peak that can be done without glacier travel is Iliniza Norte (maybe Sangay too, but it's a dangerous climb). Most of the peaks below 5000 meters can be done without glacier travel.
Check the SP pages on all these mountains for more information.