If your single member LLC does not have any employees then your accountant is correct. Note that the owner of a single member LLC can not be an employee of the business they own.
So if you have no employees you can pay at www.irs.gov/payments each quarter, and do so under the SSN of the business owner. Otherwise, with employees you must go the 941 route. Now for the 941 depending on the income of your business, you "could" be required to file the 941 monthly. But for 90% of small businesses with employees, they don't come anywhere near that requirement and are just fine filing/paying quarterly.
Note that if your state taxes personal income, the filing requirements for state taxes could be different from federal taxes. So don't let that get past you or the fines and penalties will be a real killer at tax filing time.
