You should not be needing to mess with MTU settings for RDP servers in a local LAN environment,
It's not beyond the realm of possibility as an issue, as the 3560's used a 1500 byte MTU, while all Meraki switches support jumbo frames (around 9000-byte MTUs).
The MTU is negotiated between the client and the server and the smallest of the two is selected. It is highly unlikely your clients have an MTU of anything other than 1500. So configuring a higher MTU on the server is likely to have zero impact. Configuring a lower MTU will result in a small MTU overall, but there is not a high probably of this altering your problem.
The only recent change you mention was the switches. Have you double-checked the port connections to the server are operating at full-duplex, and that there are no Ethernet errors to speak of? Ideally, configure everything for auto/auto.
Have you double-checked the connections to the clients, to make sure they are auto/auto and negotiating a full-duplex connection, and that there are no Ethernet errors reporting on the MS switch port?
Anything in the Meraki event log for switches? Any spanning tree issues reported, port problems, or anything else interesting? Duplicate IP address warnings?
If you test it off one of your thin client devices and a Windows computer - are both clients equally affected?
Are there any firewalls in between the users and the servers? If so, is it reporting anything interesting?
If all else fails, you are going to need to take some packet captures at the server and client end to understand what they are seeing.