Okay, I am having difficulty imagining your network. Here's what I would do.
1. Go to the end user's computer/server initiating the connection - the 10.x.100.0 network device?
2. Find the default gateway for that device.
3. Log into the gateway (should be the DHCP relay/router) possibly your 2960, but I'm not sure with the information provided.
4. Check the routing table(like you just did). if the 192.168.x.0 network is directly connected (2960 is your default gateway), then follow the layer 2 path to the end device.
4.1. If the VLAN/layer 2 connections if not directly connected, follow static route/routing table to get to the 192.168.x.0 network.
5. Check the collisions and port statistics of every port along the path.
If it doesn't touch your firewall, then it doesn't touch your firewall, but also check the path that the packets take back. Sometimes it is different and that could be your problem (Split horizon). Since you aren't using a routing protocol, your path will never change/bounce so it should make it easy to trace. It appears to me that routing is not your problem. Basically the only information that I can't tell from what is here is the default gateway of the end devices., and logically how your firewall connects to either the ASA or your 2960s..
This sounds like a OSI layer 1 or 2 problem to me. Unless the firewall, or an ACL, is blocking a specific port and your applications use multiple ports? Or, your network is saturated and 802.1q (QoS) is being the problem...
There's a few possilities I guess. Hope I didn't confuse you with my rambling.
Good luck!