So this happened a few years ago in a college town in California. A police officer came to the door of the house I was living in, and told us that someone had stolen a large theta from the front of a frat house, and that someone walking down our street at around 6:00am had seen the letter on the lawn in our front yard. Given that it was stolen property the police officer told us that he could get a search warrant for our house, and if we didn't let him in he would block our cars in with his car and wait for the search warrant. We had nothing to do with the stealing of the letter and woke up long after it was gone. We assumed someone had drunkenly stolen it and dropped it on our front lawn. One of my roommates ended up getting scared and letting the officer in so the point was moot, but I did not think the officer was being truthful in saying he could obtain a search warrant. The stolen property had been seen on our lawn, but there was no way anyone could prove (because it was false) that we had anything to do with stealing the letter, or that the letter was even still around.
Keep in mind that a warrant doesn't require proof that you stole the property or have the property. Instead, the warrant is just authorization to look for that proof.
The standard for securing a warrant is probable cause, which is a much lower bar to clear than people seem to think. It just requires that given everything the officer knows, there's a "substantial probability" that a piece of evidence will be in a given place at a given time.
If the officer swore to a judge that a reliable source had told him that the letter was stolen, then seen in your front yard, and not seen since, I wouldn't be at all surprised if the judge gave him a search warrant.
The search warrant is there to get evidence against whoever committed a crime, not (necessarily) against you. If a bankrobber flees and throws his gun over your garden fence, then the police should be allowed to get the evidence against the bank robber from your garden, even though it is extremely unlikely that you committed any crime.
On the other hand, what if they search without a warrant: The police would be violating your rights. Any evidence against you that they find wouldn't be admissable in court. However, since you were not the thief, or the bankrobber, they wouldn't have violated the rights of the thief or the bankrobber, so evidence against them would be admissable (assuming the guilty party wasn't you, or living with you, or your guest).
PS. I remember hearing of cases where a murder was committed, years later the police figured out the victim might have been buried in some person's garden, and the police easily got a search warrant to dig up the garden of a new home owner who was 99.9999% not guilty of any crime. Because there was a very good chance of finding criminal evidence against the previous home owner. I'd be curious if the police has to pay for damage to your garden, or your home owner's insurance would have to pay, or if this is just tough luck.