+1 to all of the above. The below is more clarification/edification on the reasoning for case fans being the number one choice if anyone wants to read or this thread pops up in years to come.
Regardless, temperature monitoring should always be used to guide performance of any modifications chosen. tlw
Though air-cooled engines can get very hot indeed if run without a moving airflow passing over them.
Anyone who has ridden an air-cooled bike hard and then stopped at an "epic" red light can attest to this. I should have clarified that a bit (the point I really wanted to make is what tlw said above), since when in motion it is fine and dandy, but when stopped the rider is sitting on a ""slow cooker" which is only getting heat transfer from natural convention.
[more of an aside here] Just to clarify (and throw a little engineering in for fun), heat transfer is driven by delta T (difference from heat source to heat sink), with the mass flow rate and specific heat capacity of the fluid proportional to the heat transferred (typically written as q=mCΔT, or variations thereof). For fluid-flow heat transfer, air is considered a "fluid."
Practical application everyone can attest to is "wind chill" - going into a cold environment is not so bad with no air flow, but in a windy environment heat transfer is noticeable immediately, driven also because the "laminar layer" ("dead air") near the skin's surface gets "thinner" without warm clothes to trap it there. An important point with this is "wind chill" cannot make anything colder than the delta T; it just jacks up the heat transfer rate. [/aside]
Both of the above fit into the recommendation for case fan(s). The case is not only a protective cover, but also forces air flow in a specific direction, pulling air from the box near the CPU/PSU and forcing it into the room (hopefully) on the other side of the case. Without the "constraint" of maintaining cold flow over the heat source (maintaining delta T at a good value), you run into "natural convection" which is more vertical like how a chimney works (or the example of sitting at a red light on a motorbike). The case fan also helps to stir the heat coming out so that it dissipates quicker when it enters the room - depending on flow restraints outside the box it is also possible to pull the heat right back into the front (eventually).
Temperature monitoring is highly recommended if you alter "intended design," which may be fine for your situation, but the forced air flow from a case fan does ensure a better heat transfer inside the box (and lower fan speeds all around) - essentially "wind chill" for your CPU, which in this case is a good thing.