The Sunday blasts destroyed the main bridge over the Choloki river and a smaller one near the village of Kakuti.
Rebels hope to slow down any government attempt at reintegrating the region back into the ex-Soviet Black Sea state.
Ajarian leader Aslan Abashidze, interviewed by Russia's Itar-Tass news agency, said the demolitions were intended to rule out any government troop deployment south.
"According to our data, some units of the Georgian armed forces taking part in the manoeuvres ... set up tents only a kilometre from the border," Abashidze was quoted as saying in the region's main town Batumi.
Georgian officials denounced the explosions and said the army had no intention of moving against Adzhara.
"Ajaria has totally isolated itself," Ivan Merabishvili, head of Georgia's Security Council, told Rustavi-2 television.
Bordering Turkey, Ajaria is the
southernmost rebellious region
"This shows that Aslan Abashidze and his regime is in their final agony. They are very frightened."
Ajaria is one of three Georgian regions operating beyond the control of central government but, unlike the other two, has never declared itself independent.
Abashidze runs the region, presiding over his own armed forces and declining to pay taxes to the state budget.
Georgia and Ajaria came close to military confrontation in March when Saakashvili was prevented from entering the region during an election campaign. Both sides put forces on alert.
Backed by Washington, Saakashvili was elected in January after leading a bloodless revolution that ousted veteran leader Eduard Shevardnadze.
He has called for the Ajaria leader's removal, but vows to use only peaceful means.