I recently replaced my kitchen counter tops and sinks. The previous sink/counter top had a hole for a "spillover" from the dishwasher and the garbage disposal. The new one does not. However, the plumbing under the sink was not modified for this (unbeknownst to me). I was wondering why I would occasionally see water under the sink, but last night, I heard it and witnessed water - lots of water! - coming from that overflow that goes no where now! The guy who rigged it up is a neighbor that I don't want to engage again. Any suggestions on how to fix this without having to cut through the new counter top?

What you refer to is called an air gap. Water is pumped up from the dishwasher to the air gap. From there it flows down from gravity to either the drain or the disposal. This prevents contaminated water in the drain from flowing back into the dishwasher. The air gap is mounted on the rim of the sink, not the counter top. If the hose between the air gap and the drain gets stopped up water flows out the air gap and into the sink. Some sinks come with a hole in the rim for mounting an air gap, some do not. Sometimes the hole for the air gap is used to mount a soap dispenser. Additional holes can be drilled in stainless steel for mounting air gaps or soap dispensers.

What is called a "high loop" can be used instead of an air gap. That is when the dishwasher drain hose is attached to the highest point underneath the counter top. Water from the dishwasher is pumped up to that point and then flows down to either the disposal or to the drain of the sink. This prevents contaminated water from the drain flowing into the dishwasher.
Some localities require an air gap. Some permit a high loop. You must have one or the other. If you don't the waste water from the sink will flow down and into the dishwasher.