we are selling our house in Costa Rica and the buyer will be depositing $350,000.00 into our account in Oregon. Can the IRS reporting be avoided if the deposit is a check rather than cash? The money would most likely come from within the US and not from out of country.

Cash transactions are reportable. That doesn't mean the IRS uses this data to monitor your income taxes. It may or may not be a trigger to them. In any case you are expected to properly complete your tax returns and there may be reporting elsewhere of the transaction. Trying to conceal income and always being in fear that the IRS will seize your deposit accounts is not a way to live.

We also recommend notifying your bank that they will be receiving a large deposit. It allows them to plan better, especially if you'll want to move those funds out any time soon. This advise is repeated if you intend on depositing or withdrawing that much cash.