Try putting in a SD card of a size that this camcorder supports. Why would JVC replace it? It's like asking Ford to replace your tires. That is, the most costly route?
http://support.jvc.com/consumer/products/glossary.jsp?gId=344 notes other software than Picasa so more questions from me. Why didn't you install the same software you had before to get it to work?
It's possible the camera has failed or your new PC is missing some needed driver which is either on said JVC software or something the computer maker forgot. I can't comment as we only know what you tell here.