Well since insurance operates on a principle of indemnity you should actually pay up front and then claim for reimbursement - that's just the way it works in theory (and sometimes in practice too). Usually if you have larger bills or expect them it would be a good idea to get in touch with your insurer as soon as you can, insurance companies are normally reasonable and they don't much like nasty surprises in the form of unexpectedly high claims.

IN our own experience we had claims for some tens of thousands of CHF and in that case the doctors/clinics contacted the insurer direct and settled everything between themselves - this worked very smoothly.