Just because a fan is connected to a 3 pin fan does NOT mean it is an uncontrolled 100% "on" configuration. There are few "system" fans that are 100% on. Most are 3 pin voltage controlled through the BIOS and are thermally regulated. 4 pin PWM motherboard connections are preferred, but 3 pin voltage controlled headers are just fine and in most cases are not the "sys" type, but are the "cha" or chassis type.
Personally, I rarely recommend using the integrated fan controllers that come with cases. They are usually low quality affairs with a tendency to overdraw current and fry things. The specs for your motherboard list one 4 pin CPU fan header and three 4 pin chassis fan headers. That means six case fans plus the CPU cooler could be controlled via your motherboard using three of these for the case fans which work equally well using 3 or 4 pin fan models as the fourth pin is only for the PWM signal and is not essential when using 3 pin voltage controlled fans.
http://www.newegg.com/Product/Product.aspx?Item=N82E16812162026
Two fans per header is well within tolerance. Three fans per header can work, but it's not recommended as you move dangerously close to overloading the circuit design by doing so. Two per header won't even come close to the amperage limit. This would allow all your case fans to be voltage or PWM controlled through the motherboard BIOS, and most your fans should be compatible. There are likely standard 3 or 4 pin to Molex adapters being employed in order to connect to the power supply and if so they can be removed in order to use the standard fan connectors that are directly integrated with the fan cabling.
This would make your system far more efficient, no need to manually adjust fan speeds, use far less power saving money as they won't be running at high speeds unless demanded by the current load. The system will be far quieter the majority of the time and you will have a system that is close to a neutral air pressure with them all being controlled thermally, neither positive nor negative, which is a good balance of performance and dust suppression. Dust suppression is the only real advantage of a positive pressure configuration.