I can think of at least 3 categories of problems to check, as first steps:
1) New Hardware configuration:
Memory utilization -- what % free memory is available? Low memory also causes high disk usage as the OS thrashes memory pages to/from disk.
Regardless of memory, if your game/app writes to disk, check your disk wait during one of the CPU blips.
2) New software configuration environment -- "Configuration Secrets" that exist on your old box that need to be re-applied to the new server:
Check your virus-scanner config -- often there are files that need to be exempted from the 'scan-before-access' protection on a server. Were there configs like this on the previous server that need to be re-applied in the new environment?
What about things like CPU affinity?
There could be anything in this category; it's why configuration management is so important. Hopefully one of the other two categories holds the real problem.
3) New Network environment:
Is the new host in a different datacenter? If your server communicates to a backend service (authentication/database/etc) of any kind, verify that throughput between your server and that service are sufficient.
A bad router port/NIC could also be an issue; but that should be easier to detect -- run a continuous ping and pay attention to the delay times during the CPU drop.