Assume you can be coerced into encrypting the same plaintext three times, under three different public keys. You can; it's happened.
Then an attacker can trivially decrypt your message, by:
The CRT says you can take any number and represent it as the combination of a series of residues mod a series of moduli. In the three-residue case, you have:
result = (c_0 * m_s_0 * invmod(m_s_0, n_0)) + (c_1 * m_s_1 * invmod(m_s_1, n_1)) + (c_2 * m_s_2 * invmod(m_s_2, n_2)) mod N_012
c_0, c_1, c_2 are the three respective residues mod n_0, n_1, n_2 m_s_n (for n in 0, 1, 2) are the product of the moduli EXCEPT n_n --- ie, m_s_1 is n_0 * n_2 N_012 is the product of all three moduli
To decrypt RSA using a simple cube root, leave off the final modulus operation; just take the raw accumulated result and cube-root it.