Sure. The Wii comes with composite video and l/r audio connections. So any TV with those connections would work.

Quality-wise, using the composite video depends on the quality of the 'comb filter' in the display. Look for a TV that has a 'digital 3D' comb filter. It is likely that your older TV has a better comb filter than your newer TV. (Or possibly you need to enable the comb filter via a menu option?)

The wii supports s-video. So if your set has s-video connections, the comb filter is bypassed by purchasing a new cable set. S-video cables for the wii cost almost nothing on ebay.

On my 10 year old Panasonic TV, which has both a 3D comb filter for composite video, along with S-video inputs, there is little to no difference in picture quality between comb-filtered composite and s-video. Turn the comb filter off, and there are very noticable differences between composite and s-video.

However, does your newer TV have 'component video' (separate red/green/blue) connections? If so, it is very worthwhile to buy a set of component video cables for it. Again, you can get them on ebay for next to nothing. And if the newer set supports High Definition, the component cables allow the use of 480p progressive scan. (Be sure to select it in the setup menu.) This is the best you can do with the wii.