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It would not be able to pass close enough to Mars AND Venus, unless there is some way for Jesus to make the earth take some roundabout path.
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If Mars and Venus were opposite the Sun from Earth, and Earth were nudged further inside its orbit, as it falls past the sun, Earth could easily pass near both Venus and Mars. If the timing were right, Earth could pass close enough to pick up acceleration from both of them, sending it to the outer solar system, and likely altering the orbits of Mars and Venus as it does so.
It is, of course, wildly unlikely. I just wanted to point out that it's possible for Earth to pass both Mars and Venus without looping or doubling back. Shoot, you could probably even fit Mercury into the mix; though it's effect on Earth would be negligible, given the proximity of the Sun, Earth could have quite the effect on Mercury.
Personally, I think we should pile up the gas giants, big balls of fuel that they are, then fashion the rocky planets and asteroids into a vessel that can ride within the fuel ball. When this Sun doesn't want us, anymore, we'll just take our planets and go.